IS 6200 (Part 1) :2003 Indian Standard STATISTICAL TESTS OF SIGNIFICANCE PART 1 NORMAL, ( t - AND F- TESTS Third Revision ) ICS 03.120.30 0 BIS 2003 BUREAU MANAK OF BHAVAN, INDIAN STANDARDS ZAFAR MARG 9 BAHADUR SHAH NEW DELHI 110002 Jtine 2003 Price Group 8 Statistical Method for Quality and Reliability Sectional Committee, MSD 3 FOREWORD This Indian Standard (Part 1) (Third Revision) was adopted by the Bureau of Indian Standards, after the draft finalized by the Statistical Method for Quality and Reliability Sectional Committee had been approved by the Management and Systems Division Council. This standard was originally published in 197I and covered the industrial applications of three main tests of significance, namely, t-test, F-test and X2-test. It was then revised in 1977 into four parts to include tests for normality as also some non-parametric tests, which have wide application in industry. The second revision of this part had been taken up in 1995 to rearrange the contents of the standard by putting together tests of the same hypothesis under different assumptions and requiring different use of test statistics. In addition, the revised standard includes: (a) the basic concepts of formation of null and alternative hypothesis; (b) Fisher-Behren's test for testing equality of means of two populations when the variances are not known and not assumed to be equal; and (c) examples on testing for proportions. The third revision of the standard has been undertaken to: a) b) c) modify the examples so as to make them practical and also that the reported test results to be of same decimal places as actually obtained in practice, include the table for finding (3from tan O to be used in Fisher-Behren's test, and incorporate many editorial corrections. The tests of significance described in this standard are useful in many problems of industrial experimentation. However, they should not be generally used for lot acceptance purposes for which IS 2500 (Part 1) :2001 `Sampling inspection procedures : Part 1 Attribute sampling plans indexed by acceptable quality level (AQL) for lot-by-lot inspection' and IS 2500 (Part 2): 1965 `Sampling inspection procedures : Part 2 Inspection by variables for percent defective' may be referred to. In addition to this Part 1, IS 6200 has following three parts: Part 2 X2-test Part 3 Tests for normality Part 4 Non-parametric tests The composition of the Committee responsible for the formulation of this standard is given in Annex F. IS 6200 (Part 1) :2003 Indian Standard STATISTICAL TESTS OF SIGNIFICANCE PART 1 NORMAL, f - AND F- TESTS (Third 1 SCOPE I Revision ) in decision-making. They are extreme] y useful in finding out whether, in the case of one population, the mean value differs significantly from certain specified value or whether, in the case of two populations, the mean values differ significantly from each other. Thus, it may be desirable to find out whether a new germicide is more effective in treating a certain type of infection than a standard germicide, whether a new method of sealing light bulbs will increase their life or whether one method of preserving foods is better than another in so far as the retention of vitamins is concerned. In such cases, it would be necessary to examine whether the mean values obtained can be deemed as same or different. There may also be cases where it maybe worthwhile to find out whether one inspector is more consistent than another or whether a new source of raw material has resulted in a change in the variability of the output or whether the temperature of the bath in which the cocoons are cooked affects the uniformity of the quality of silk. In these cases it will be necessary to determine whether the variances obtained are the same or not. 4.2 Formulation of Hypothesis 1.1 This standard (Part 1) lays down the following tests of significance: a) One-sampIe test -- Testing of mean of a population against a specified value; i) b) when the population variance is known, and ii) when the population variance is not known. Two-samples test -- Testing equality of means of two populations; i) when the variances are known, and ii) when the variances are not known. c) Testing for equality of variances of two populations; and d) Testing for proportions. 1.2 This standard does not include the t-tests for regression coefficient and correlation coefficient as they are covered in IS 7300: 1995 `Methods of regression and correlation (@w revision)'. 2 REFERENCES The following standards contain provisions, which through reference in this text constitute provisions of this standard. At the time of publication, the editions indicated were valid. All standards are subject to revision and parties to agreements based on this standard are encouraged to investigate the possibility of applying the most recent editions of the standards indicated below: 1S No. 7920 Title For taking a decision using statistical tests of significance, the first step is to form the hypotheses, namely, Null hypothesis (110) and alternative hypothesis (l-Zl). 4.2.1 Null Hypothesis (HO) The procedure commonly used is to first setup a null hypothesis regarding equivalence (no difference) of the assumed population mean and the specified value. The question on which the decision is called for, by applying the tests of significance, is translated in terms of null hypothesis in such a way that this null hypothesis would likely be rejected if there is enough evidence against it as seen from the data in the sample. For example, in the case of new germicide, a null hypothesis will be that it is not more effective than a standard germicide; or in the case of light bulbs the new method of sealing does not increase the life of the bulb. 4.2.2 Alternative Hypothesis (H,) (Part 1) :1994 (Part 2) :1994 Statistical vocabulary and symbols: Probability and general statistical terms (second revision) Statistical quality control (second revision) 3 TERMINOLOGY For the purpose of this standard, the definitions given in IS 7920 (Part 1) and IS 7920 (Part 2) shall apply. 4 BASIC CONCEPTS 4.1 Statistical tests of significance are important tools Alternative hypothesis is opposite to the null hypothesis. It maybe two-sided or one-sided. 1 IS 6200 (Part 1) :2003 4.2.2.1 Two-sided hypothesis In some situations, it may be of interest to find out whether lot mean differs significantly from a specified value irrespective of the fact that this difference is positive or negative. For example, in the manufacture of certain cylindrical rods, one may have to examine whether average diameter differs significant] y from the specified nominal value or one may wish to determine whether the night shift production differs from that of the day shift in respect of certain quality characteristics of the item. In such cases the test is said to be two-sided. 4.2.2.2 One-sided hypothesis In some other situations, only the positive difference or the negative difference between the lot mean and specified value may be of interest. For example, when drinking water is tested for bacteria count, only high values of count maybe the source of concern or when cases of concrete are tested for strength, low values may have to be detected. In such cases, the test is said to be one-sided. 4.3 Level of Significance 4.3.1 There are two kinds of errors involved in taking the tabulated value. If the calculated value is greater than or equal to the tabulated value of the statistic, then HO is rejected, thereby accepting H,; otherwise HOis not rejected. For practical purpose, Ho not rejected is taken as, if it is accepted. 4.5 For each test of significance, certain underlying assumptions are made (see 5.1 and 8.2). Hence, it is important that these tests are not used indiscriminately. If the assumptions are in doubt, it is advisable to obtain the guidance of a competent statistician to ascertain the feasibility of application of these tests. 5 ONE SAMPLE TEST -- TESTING OF A POPULATION VALUE AGAINST OF MEAN A SPECIFIED , 5.1 To judge"whether the population mean ~) differs significantly from a specified value, ~0, a sample of size n is taken from the population and sample mean E is calculated. In this case, null hypothesis is: HO : p = AO Depending upon the situation, any one of the following three alternative hypotheses may be selected: a) b) H, : p# ,uO (two-sided), H, : A > PO(one-sided), and the decision based on the tests of significance, namely: a) Type Z error -- Error in deciding that a C) HI : p <#0 (one-sided). significant difference exists when there is no real difference. Type II error -- Error in deciding that no difference exists when there is a real difference. It is assumed that the observations follow normal distribution and are drawn at random. Depending upon the knowledge about the population variance, two cases may arise as given below. 5.2 Population Variance (oz) is Known In this case the normal test is applied by computing the statistic z= I y ­pOIJn/O z=(X ­@4n/o b) 4.3.2 Type I error and Type H error is also called error of the first kind and error of the second kind respectively. This process of decision making is given below: z=@~ X)dn/o forlZ1:##pO forlfl:p>~o forlll:pffO for H1:p Pz (one-sided), and C) HI : P, c Pz (one-sided). Depending upon the knowledge about the variances of the two populations, two cases may arise as given below. 6.2 Population Variances are Known In this case the normal test is applied by computing the statistic, z = Iz ­ ~ 1/[(02,/n, ) + (G22/n2)]%for H1 : PI #,u2 z = ( Y ­ ~)/[(cr2,/n,) + (cr22/n2)]%for H, : p, > ,U2 z = ( ~ ­ Y )/[(62,/ n,) + (622/n2)]% for H, : K1 < A2 Solution 2.0 2.1 1.9 2.0 1.6 2.9 From the data, the average breaking load for the two manufacturers is obtained as: Manufacturer A: Y =2.17 Manufacturer B: ~ = 2.52 The null hypothesis (HO) is that the mean breaking load of the yam for the two manufacturers is the same. a) In this case, the alternative hypothesis is that the mean breaking load of the yam for the two manufacturers is different. The teststatistic is computed as, z = Iy ­J l/[(cr,2/n,) + (crz2/n2)]'" = 12.17- 2.521/[(0.36)2/10) + (0.36)2/12)]"2 Where o, and 02 denote the known standard deviations for the two populations. Y and J are the means of the samples of sizes n, and nz drawn from these populations respectively. The table values ofz are given in Annex A. These values will be used for taking the decisions as per 4.4. 6.2.1 Example 4 Ten samples of a particular type of yam were tested from the consignment of manufacturer A whereas twelve sample results were available for manufacturer B. The breaking load (in Newton) of the samples tested is given in Table 1. It is known that the standard deviations for breakifig load have been satisfactorily established from the earl ier measurements as 0.36 for both the manufacturers. It is desired to compare whether the mean breaking load of the yam for: a) b) two manufacturers is significantly different from each other. manufacturer A is less than that for manufacturer B. b) = 2.59 Since this calculated value is greater than the tabulated value of 1.960 at 5 percent level of significance (for two-sided test) from Annex A, the null hypothesis, that the mean breaking load of the yarn for the two manufacturers is same, is rejected at 5 percent level of significance. In thk case, the alternative hypothesis is that the mean breaking load of the yarn for manufacturer A is lower than that of manufacturer B. The test statistic is computed as z = ( J ­Z )/[(62,/n,) + (c22/n2)]y1= 2.59 Since this is greater than 1.645 and 2.326, the tabulated values at 5 percent and 1 percent levels of significance respectively, for one-sided test (see Annex A), the null hypothesis that the mean breaking load of the yam for the two manufacturers is same, is rejected at 5 percent as well as at 1 percent level of significance. 6.3 Population Variances are not Known 6.3.1 Independent Populations Assumed to be Equal with Variances In this case, t-test is applied by computing 4 the IS 6200 (Part 1) :2003 following statistic: t= I x ­ J I/s'[(n, + n2)/n,rr2]% for~, : Al #p2 t= (x ­ J)/s'[(n, + rr2)/n,nJK for~l : p, >fl~ t= (y ­ J)/s'[(n, + n2)/n,r22]% for~, : fll pO The tabulated values ofz are given in Annex A. These values will be used in taking the decision as per 4.4. 7.1.1 Example 7 A purchaser picks up at random a sample of size 15 from a lot and finds one non-conforming item. It is required to test the hypothesis that the lot contains 5 percent non-conforming items. Here, 7 = 1/15= 0.067; n = 15;pO= 5/100= 0.05 To examine if the proportion non-conforming in the lot, PO is 0.05, the null hypothesis Ho: n = PO is tested against the alternative hypothesis lfl: x #P. The z statistic is calculated as: = 10.067- 0.051/[0.05 (1 - 0.05)/15]fi = 0.017/0.056= 0.30 This value is less than 1.96, the tabulated value of z at 5 percent level of significance (two-sided) (see Annex A). Hence, the null hypothesis that the proportion non-conforming in the lot is 0.05 is not rejected. 7.2 In the case of testing the equality of two proportions corresponding to two populations, ifpl and p2 are the two proportions obtained on the basis of samples of size nl and n2 drawn from the two populations respectively, then calculate: P' = (~p, + %P*)@l + ~') Compressive Strength for Sand (;) 24.1 23.2 26.0 23.7 26.5 27.0 27.1 23.5 26.1 25.3 26.3 23.3 22.7 25.8 27.5 22.2 Differences, d= A-B (2) - (3) (4) 0.1 -0.2 0.0 -43.2 0.0 0.0 0.2 4.4 0.1 -0.2 -0.2 0.0 43.3 0.0 0.0 -0.1 (?) 24.0 23.4 26.0 23.9 26.5 27.0 26.9 23.9 26.0 25.5 26.5 23.3 23.0 25.8 27.5 22.3 To examine if the average difference is significantly different from zero, the null hypothesis is HO: J = O against the alternative hypothesis, Hl: d # O. t= \j ­Oldn/s=O.08 X~16/O.16=2.O This value is less than 2.131, the tabulated value of tat 5 percent level of significance (two-sided) with 15 degrees of freedom (see Annex B). Hence the null hypothesis that the average compressive strength of the cement does not differ with the two grades of the standard sand is not rejected. 6.3.3.2 If the number of observations in the sample, n, becomes large, the t-test becomes equivalent to the Normal test. When the sample size is greater than 30, Normal test may be used instead of t-test for all practical purposes. 7 TESTING FOR PROPORTION 7.1 The normal test described in 5.2 and 6.2 may also In this case, the null hypothesis is HO: Z, = Z2 against the alternative hypothesis H, : Z, # Z2 or Hi : Xl < zz or HI : n, > Zz. The z-statistic to be used is obtained as: z = I~ , ­ ~ `1/~'(1 ­p') (n, + n2)@JX for Hl : nl#pz z = ( ~ , ­ ~ `)/~'(1 ­p') (n, + n2jh,n2]fi for H, : ml> xl z = ( ~ *­ ~ J@'(1 ­p') (n, + nz~n,n2]% for Hl : xl< X2 be extended to the testing for proportions. If the binomial proportion p (obtained as the ratio of the number of successes in n repetitive trials) is to be tested against a specified value pO then the null hypothesis is HO: n = PO against the three possible alternative hypotheses, namely, H,: n #p, (two-sided) H,: n PO(one-sided) where n is the population proportion of success. Further, 6 The rules for rejecting or not rejecting hypothesis are similar to those in 7.1. 7.2.1 Example 8 the null A machine puts out 20 imperfect articles in a sample of 500. After machine is overhauled, it produced 3 imperfect articles in a batch of 100. Has the machine improved? Here, n,= 500, ~,= n2= 20/500 = 0.04, 100, ~z = 3/100 = 0.03 IS 6200 (Part 1): 2003 The null hypothesis is HO:nl =X2 and the alternative hypothesis is HI : X, > Xz p' = (500 X 0.04+ 100 x 0.03)/(500+ The test statistic is computed as: z = (p ~­ p Jb'(1 It is required to find whether any one method gives more consistent results: Solution 100)= 0.038 ­p') (n, +fz2)/n,n2]~ = (0.04 - 0.03)/[0.038 (1 - 0.038) (500 + 100)/500x loo]%= 0.01/0.021 = 0.48 The variances of the two sets of results as indicated in 8.1 are obtained as SI2 = 0.062 for Method X, and z = ().()19for Method Y. S2 Table 4 Chemical Determination by Two Methods (Clause 8.1.1) Method X (1) 1.00 0.95 1.45 0.70 1.05 0.95 1.15 0.95 1.35 1.10 1.50 0.80 0.95 1.10 0.85 1.30 0.75 0.90 1.55 1.25 This value is less than 1.645, the tabulated value of z at 5 percent level of significance (one-sided) (see Annex A). Hence the null hypothesis is not rejected, that is the machine has not improved. 8 TESTING EQUALITY OF VARIANCES 8.1 To test the equality of the variances of the two populations, say 612 and cr22, null hypothesis is, HO: o,*= CJ22. Method Y (2) 0.75 (J.70 0.70 0.95 0.75 0.90 1.00 1.05 1.10 0.95 0.85 0.80 0.70 1.05 0.85 The alternative hypothesis may be one-sided or twosided as: H, : cr,z # 622 (two-sided), and HI :6,2>622 (one-sided). Thus, if the variances of the samples of sizes n, and n2 from the two populations be s12andS22, respectively, the following statistic is computed: F = S,2/S22 F = S,2/S22 F = S,2/S *2 when H, : 0,= # CJ22 when H, : o,* B 022 when H, : o,*< CJ22 where S,*= Z(X­ Y)*/(n, ­ l)=(Zx?-n, z*)/(n, ­ 1) s2==z(y­ y)2/(n2­ l)=(zy­n2y=)/(n2­ 1) The larger of two variances is taken in the numerator. Ihe value of Fis associatedwith (n, ­ 1), (n2­ 1)degrees of freedom when S)*> S22. In case S22is greater than S]*, the degrees of freedom would be (nz­ 1), (n, ­ 1). The tabulated values of F are given in Annex E. These values will be used for taking the decision as per 4.4. 8.1.1 Example 9 In this case, the null hypothesis is that the variations in results of the chemical determination by the two methods are the same against the alternative hypothesis that one method gives more consistent results than the other. Applying the F-test, F = S,2!S22 = 0.062/0.019 = 3.235 There are two alternative methods X and Y available for making a chemical determination in parts per million (ppm). The results of 20 determinations by Method X on a given sample and 15 determinations by Method Yon the same sample are given in Table 4. Since the calculated value of F is greater than 2.40 which is the tabulated value at 5 percent level of significance with 19 and 14 degrees of freedom in the Table in Annex E (for one-sided test), the null hypothesis is rejected. Further, since S22is smaller than s,*, it is concluded that the Method Y gives more consistent results as compared to Method X. 8.2 Like t-test, the F-test is also based on the assumption of the normality and independence of the observations (see also 4.4). IS 6200 (Part 1) :2003 ANNEX A (Clauses 5.2,5.2.1,6.2,6.2.1, CRITICAL Significance (1] 0.05 Level VALUES OF STANDARD 7.1.1 and 7.2.1) NORMAL DISTRIBUTION (Z) (a) One- Sided Test (2] 1.645 Two-Sided Test (3) 1.960 0.01 2.326 2.576 ANNEX B (Clauses 5.3.1,5 .3.2,5.3.3,6.3.1.1 CRITICAL Degrees of Freedom One-Sided and 6:3.3.1) VALUES OF t-DISTRIBUTION Levels . Two-Sided Test Signtflcance Levels - Test Signljkance (1) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29. 30 31 5 percent (2) 6.314 2.920 2.353 2.132 2.015 1.943 1.895 1.860 1.833 1.812 1.796 1.782 1.771 1.761 1.753 1.746 1.740 1..734 1.729 1.725 1.721 1.717 1.714 1.711 1.708 1.706 1.703 1.701 1,699 1.697 1.645 1 percent (3) 31.821 6.965 4.541 3.747 3.365 3.143 2.998 2.896 2.821 2.764 2.718 2.681 2.650 2.624 2.602 2.583 2.567 2.552 2.539 2.528 2.518 2.508 2.500 2.492 2.485 2.479 2.473 2.467 2.462 2.457 2.326 8 5 percent (4) 12.706 4.303 3.182 2.776 2.571 2.447 2.365 2.306 2.262 2.228 2.201 2.179 2.160 2.145 2.131 2.120 2.110 2.101 2.093 2.086 2.080 2.074 2.069 2.064 2.060 2.056 2.052 2.048 2.045 2.042 1.960 1 percent (5) 63.657 9.925 5.841 4.604 4.032 3.707 3.499 3.355 3.250 3.169 3.106 3.055 3.012 2.977 2.947 2.921 2.898 2.878 2.861 2.845 2.831 2.819 2.807 2.797 2.787 2.779 2.771 2.763 2.756 2.750 2.756 1S 6200 (Part 1) :2003 ANNEX C (Clause 6.3.2) VALUES OF NATURAL )eg ecs 0' TANGENTS 42 ` 00.7 01222 02968 04716 064767 08221 09981 11747 13521 15302 17093 18895 20709 22536 24377 26235 28109 30001 31914 33848 35805 37787 39795 41831 43897 45995 48127 50295 52501 54748 57039 59376 61761 64199 66692 69243 71857 74538 77289 80115 83022 86014 48 ` 00.8 01396 03143 04891 06642 08397 10158 11924 13698 15481 17273 19076 20891 22719 24562 26421 28297 30192 32106 34043 36002 37986 39997 42036 44105 46206 48342 50514 52724 54975 57271 59612 62003 64446 66944 69502 72122 74810 77568 80402 83317 86318 54 ` 00.9 01571 03317 05066 06817 08573 10334 12101 13786 15660 17453 19257 21073 22903 24747 26608 28486 30382 32299 34238 36199 38186 40200 42242 44314 46418 48557 56733 52947 55203 57503 59849 62245 64693 67197 69761 72388 75082 77848 80690 83613 86623 Mean Di@-ences 45 123 29 29 29 29 29 29 29 30 30 30 30 30 30 31 31 31 32 32 32 33 33 34 34 34 35 36 36 37 38 38 39 40 41 42 43 44 45 46 47 49 50 58 58 58 58 59 59 59 59 59 60 60 60 61 61 62 63 63 64 65 66 66 67 68 69 70 71 73 74 75 77 78 79 82 84 86 88 90 92 95 98 87 87 87 88 88 88 88 89 89 90 90 91 92 93 93 94 95 96 97 98 99 10 10: 116 146 116 146 117 146 117 146 117 146 117 147 118 147 118 148 119 149 120 150 120 150 121 152 122 153 124 155 124 155 125 157 127 158 128 160 129 162 131 164 133 166 134 168 136 170 1 OO.O .00000 6" OO.1 00175 01920 12 ` 18 24 ` 00.4 00.2 00.3 00524 02269 04016 05766 07519 09277 11040 12810 14588 16376 18173 19982 21804 23639 25490 27357 29242 31147 33072 35019 36991 38988 41013 43067 45152 47270 49423 51614 53844 56117 58435 60801 63217 65688 68215 30 " 00.5 00873 02619 04366 06116 07870 09629 11394 13165 14945 16734 18534 20345 22169 24008 25862 27732 29621 31530 33460 35412 37388 39391 41421 43481 45573 47698 49858 52057 54296 56577 58905 61280 63707 66189 68728 71329 73996 76733 79544 82434 85408 '36' 00.6 01047 02793 04541 06291 08046 09805 11570 13343 15124 16914 18714 20527 22353 24193 26048 27920 29811 31722 33654 35608 37588 39593 41626 43689 45784 47912 50076 52279 54522 56808 59140 61520 63953 66440 68985 71593 74267 77010 79829 82727 85710 o 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 .01746 .03492 .05241 .06993 00349 02095 03667 03842 05416 05591 07168 07344 09101 10863 12633 14410 16196 00698 02444 04191 05941 07695 09453 11217 12988 14767 16555 18353 20164 21986 23823 25676 27545 29432 31338 33266 35216 37190 39190 41217 43274 45362 47483 49640 51835 54070 56347 58670 61040 63462 65938 68471 71066 73726 76456 79259 82141 85107 .08749 08925 .10510 10687 .12278 12456 .14054 14232 .15838 16017 ,17633 ,19438 21256 23087 24933 26795 28675 30573 32492 34433 .36397 .38386 .40403 .42447 .44523 .46631 .48773 .50953 .53171 .55431 .57735 .60086 .62487 ,64941 ,67451 70021 72654 75355 78129 80978 83910 17813 17993 19619 19801 21438 21621 23271 23455 25118 25304 26982 28864 30764 32685 34628 36595 27169 29053 30955 32878 34824 20 21 22 23 24 25 26 27 28 29 36793 38787 40606 40809 42654 42860 44732 44942 38587 46843 48989 10~ 138 173 10! 141 176 10: 143 179 10! 145 182 Ill II: Ii: 148 185 151 188 154 192 47056 49206 51173 51393 52395 53620 55659 55888 58201 60562 62973 65438 67960 30 31 32 33 34 35 36 37 38 39 $0 57968 60324 62730 65189 67705 70281 72921 75629 78410 81268 84208 11[ 157 196 12( 160200 12: 12! 164205 171 214 12( 167209 70542 70804 73189 73457 75904 76180 78692 78975 81558 81849 %4507 84806 132 176219 135 180225 139 185231 142 190237 147 195244 100 151 !01 252 9 IS 6200 (Part 1) :2003 ANNEX C (Continued) -- )egees -- 41 0' OO.O .86929 .90040 .93252 .96569 .00000 .03553 .07237 .11061 .15037 .19175 .23490 .27994 .32704 .37638 .42815 .48256 .53987 .60033 .66428 .73205 .80405 .88073 .96261 !.05030 !.14451 !.24604 ?.35585 !.47509 !.60509 .74748 .90421 .07768 .27085 .48741 .73205 .01078 .33148 .70463 .14455 .67128 6' OO.1 87236 90357 93578 96907 00350 03915 07613 11452 15443 19599 23931 28456 33187 38145 43347 48816 54576 60657 67088 73905 81150 88867 97111 05942 15432 25663 36733 48758 61874 76247 92076 09606 29139 51053 75828 04081 36623 74534 19293 72974 12 ` 00.2 87543 90674 93906 97246 00701 04279 07990 11844 15851 20024 24375 28919 33673 38653 43881 49378 55170 61283 67752 74610 81900 89667 97967 06860 16420 26730 37891 50018 63252 77761 93748 11464 31216 53393 78485 07127 40152 78673 24218 78938 18' 00.3 37852 90993 )4235 37586 )1053 34644 38369 12238 16261 20451 24820 29385 34160 39165 44418 49944 55767 61914 68419 75319 82654 90472 98828 07785 17416 27806 39058 51289 64642 79289 24 ` 00.4 88162 91313 94565 97927 01406 05010 08749 12633 16672 20879 25268 29853 34650 39679 44958 50512 56366 62548 69091 76032 83413 91282 99695 08716 18419 28891 40235 52571 66046 80833 30 ` 00.5 B8473 91633 94896 98270 01761 05378 09131 13029 17085 21310 25717 30323 35142 40195 45501 51034 56969 63185 69766 76749 84177 92098 00569 09654 19430 29984 41421 53865 67462 32391 18868 17159 37594 60588 B6671 16530 51071 91516 39552 97576 36 ` 00.6 88784 91955 95229 98613 02117 05747 09514 13428 17500 21742 26169 30795 35637 40714 46046 51658 57575 63826 70446 77471 84946 92920 :.01449 10600 20449 31086 42618 55170 68892 83965 .00611 19100 39771 63048 89474 19756 54826 95945 44857 .04051 42 ` 00.7 89097 92277 95562 98958 02474 06117 09899 13828 17916 22176 26622 31269 36134 41235 46595 52235 58184 64471 71129 78198 85720 93746 2.02335 11552 21475 32197 43825 56487 70335 85556 1.02372 21063 41973 65538 92316 23030 58641 5.00451 50264 5.10664 48 ` 00.8 89410 92601 95897 99304 02832 06489 10285 14229 18334 22612 27077 31745 36633 41759 47146 52816 58797 65120 71817 78929 86500 94579 !.03227 12511 22510 33317 45043 57815 71792 87161 .04152 23048 44202 68061 95196 26352 62518 .05037 55777 .17419 54 ` 00.9 89725 92926 96232 99652 03192 06862 10672 14632 18754 23050 27535 32224 37134 42286 47700 53400 59414 65772 72509 79665 87283 95417 ?.04125 13477 23553 34447 46270 59156 73263 88783 `.05950 25055 46458 70616 98117 29724 66458 `.09704 61397 }.24321 Mean Di~~erences 1 23145 52 103 155 207 259 S3 107 160 214 268 55 $2 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 -- 111 165 221 276 57 114 171 229 286 58 118 177 237296 S1 53 56 59 72 123 127 132 138 143 184 245307 191 255319 199 265331 207 276344 216 288359 300375 314392 329411 345431 362453 382477 403504 426533 452 564 481 600 511 639 546683 584731 629786 677846 75 150225 78 157235 %2 164247 %6 172259 ?1 181 272 15 191 286 00201 302 06213 319 13 226339 20240360 28255383 36273409 46292438 57314471 69338508 83366549 732915 99397596 795994 vleandifferencescause to be sufficientlyaccu. ate >5437 97144 13341 33317 55761 15240 35443 58160 B1177 83906 10216 $3735 32882 29235 13350 47374 87162 34345 B5024 91236 10 IS 6200 (Part 1) :2003 ANNEX C (Concluded) 3eg"ees 0' OO.O 6.31375 7.11537 8.14435 9.51436 11.430 14.301 19.081 28.636 57.290 - 6' 00.1 38587 20661 26356 12 ` 00.2 45961 30018 38625 18' 00.3 53503 39616 51259 24 ` 00.4 61220 49465 64275 30 ` 00.5 69116 59575 `77689 io.385 12.706 16.350 22.904 38.188 114.60 36 ` 00.6 77199 69957 91520 10.579 12.996 16.832 23.859 40.917 143.24 42 ` 00.7 85475 80622 9.05789 10.78O 13.3W i7.343 24.898 44.066 190.98 48 ` 00.8 93952 91582 9.20516 10.988 13.617 17.886 26.031 47.740 286.48 54 " 00.9 7.02637 8.02848 9.35724 11.205 13.951 18.464 27.271 52.081 572.96 1 Mean Diffetwces 2345 81 82 83 84 85 86 87 88 89 90 9.6768 9.8448 10.019 10.199 11.664 11.909 12.163 12.429 14.669 15.056 15.464 15.895 19.740 20.446 21.205 22.022 30.145 31.821 33.694 35.801 63.657 71.615 81.847 95.489 11 IS 6200 (Part 1) :2003 ANNEX D (Ckuue 6.3.2) SIGNIFICANCE OF DIFFERENCE BETWEEN TWO MEANS (P.V. Sukhatme) nl 0° 2.447 2.447 2.447 2.447 2.447 2.306 2.306 2.306 2.306 2.306 2.179 2.179 2.179 2.179 2.179 2.064 2.064 2.064 2.064 2.064 1.960 1.960 1.960 1.960 1.960 3.707 3.707 3.707 3.707 3.707 3.355 3.355 3.355 3.355 3.355 3.055 3.055 3.055 3.055 3.055 2.797 2.797 2.797 .2.797 2.797 2.576 2.576 2.576 2.576 2.576 1.5° 2.440 2.430 2.423 2.418 2.413 2.310 2.300 2.292 2.286 2.281 2.193 2.183 2.175 2.168 2.163 2.088 2.077 2.069 2.062 2.056 1.993 1.982 1.973 1.966 1.960 3.654 3.643 3.636 3.631 3.626 3.328 3.316 3.307 3.301 3.295 3.053 3.039 3.029 3.020 3.014 2.822 2.805 2.793 2.785 2.777 2.627 2.608 2.595 2.585 2.576 30° 2,435 2.398 2.367 2.342 2.322 2.331 2.294 2.262 2.236 2.215 2.239 2.201 2.169 2.142 2.120 2.156 2.118 2.085 2.058 2.035 2.082 2.044 2.011 1.983 1.960 45° 2.435 2.364 2.301 2.247 2.201 2.364 2.292 2.229 2.175 2.128 2.301 2.229 2.167 2.112 2.064 2.247 2.175 2.112 2.056 2.009 2.201 2.128 2.064 2.009 1.960 60° 2.435 2.331 2.239 2.156 2.082 2.398 2.294 2.201 2.118 2.044 2.367 2.262 2.169 2.085 2.011 2.342 2.236 2.142 2.058 1.983 2.322 2.215 2.120 2.035 1.960 3.557 3.307 3.104 2.938 2.804 3.495 3.239 3.032 2.862 2.723 3.453 3.192 2.978 2.803 2.661 3.424 3.158 2.938 2.759 2.613 3.402 3.132 2.909 2.726 2.576 75" 2.440 2.310 2.193 2.088 1.993 2.430 2.300 2.183 2.077 1.982 2.423 2.292 2.175 2.069 1.973 2.418 2.286 2.168 2.062 1.966 2.413 2.281 2.163 2.056 1.960 3.654 3.328 3.053 2.822 2.627 3.643 3.316 3.039 2.805 2.608 3.636 3.307 3.029 2.7.93 2.595 3.631 3.301 3.020 2.785 2.585 3.626 3.295 3.014 2.777 2.576 90° 2.447 2.306 2.179 2.064 1.960 2.447 2.306 2.179 2.064 1.960 2.447 2.306 2.179 2.064 1.960 2.447 2.306 2.179 2.064 1.960 2.447 2.306 2.179 2.064 1.960 3.707 3.355 3.055 2.797 2.576 3.707 3.355 3.055 2.797 2.576 3.707 3.355 3.055 2.797 2.576 3.707 3.355 3.055 2.797 2.576 3.707 3.355 3.055 2.797 2.576 5 percent points n~=6 6 8 12 24 m 6 8 12 24 w 6 8 12 24 co 6 8 12 24 w 6 8 12 24 m n* =8 n2 = 24 t22 = `== 1 percent points nT=6 6 8 12 24 00 n1=8 6 8 12 24 co 6 8 /12= 12 IIZ= 24 nz=w 12 24 w 6 8 12 24 00 6 8 12 24 w 3.557 3.495 3.453 3.424 3.402 3.307 3.239 3.192 3.158 3.132 3.104 3.032 2.978 2.938 2.909 2.938 2.862 2.803 2.759 2.726 2.804 2.723 2.661 2.613 2.576 3.514 3.363 3.246 3.158 3.093 3.363 3.206 3.083 2.988 2.916 3.246 3.083 2.954 2.853 2.775 3.158 2.988 2.853 2.747 2.664 3.093 2.916 2.775 2.664 2.576 This table has been taken from STATISTICAL TABLES by Sir Ronald A. Fisher and Frank Yates, 1953. 12 ANNEX E 8.1) CRITICAL VALUES OF THE F-DISTRIBUTION Significance Level 0.005 (Clause (For one-sided test) Degrees of Freedom (n, -I)+ (rrz-1) 12 13 14 15 16 1' 18 19 20 30 40 50 60 80 100 m * 2 3 4 5 6 7 8 9 `0 1' + 1 ; 4 5 6 : 161 18.5 10. I `.'l 6.61 5.99 5.59 5.32 5.12 4.96 4.84 4.75 4.67 4.60 4.54 4.49 4.45 4.41 4.38 4.35 4.32 4.30 4.28 4.26 4.24 4.23 4.21 4.20 4.18 4.17 4.08 4.03 4.00 3.98 3.96 200 19.0 9.55 6.94 5.79 5.14 4.74 4.46 4.26 4.10 3.98 3.89 3.81 3.74 3.68 3.63 3.59 3.s5 3.52 3.49 3.47 3.44 3.42 3.40 3.39 3.37 3.35 3.34 3.33 3.32 3.23 3.18 3.15 3.13 3.1 I 216 19.2 9.28 6.59 5.41 4.76 4.35 4.07 3.86 3.71 3.59 3.49 3.41 3.34 3,29 3.24 3.20 3.16 3.13 3.10 3.07 3.05 3.03 3.01 2.99 2.98 2.96 2.95 2.93 2.92 2.84 2.79 2.76 2.74 2.72 225 19.2 9.12 6.39 5.19 4.53 4.12 3.84 3.63 3.48 3.36 3.26 3.18 3.11 3.06 3.01 2.96 2.93 2.90 2.87 2.84 2.82 2.80 2.78 2.76 2.74 2.73 2.71 2.70 2.69 2.61 2.56 2.53 2.50 2.49 230 19.3 9.01 6.26 5.05 4.39 3.97 3.69 3.48 3.33 3.20 3.11 3.03 2.96 2.90 2.85 2.81 2.77 2.74 2.71 2.68 2.66 2.64 2.62 2.60 2.59 2.57 2.56 2.55 2.53 2.45 2.40 2.37 2.35 2.33 234 19.3 8.94 6.16 4.95 4.28 3.87 3.58 3.37 3.22 3.09 3.00 2.92 2.85 2.79 2.74 2.70 2.66 2.63 2.60 2.57 2.55 2.53 2.51 2.49 2.47 2.46 2.45 2.43 2.42 2.34 2.29 2.25 2.23 2.21 237 19.4 8.89 6.09 4.88 4.21 3.79 3.50 3.29 3.14 3.01 2.91 2.83 2.76 2.71 2.66 2.61 2.58 2.54 2.5 I 2.49 2.46 2.44 2.42 2.40 2.39 2.37 2.36 2.35 2.33 2.25 2.20 2.17 2.14 2.13 239 19.4 8.85 6.04 4.82 4.15 3.73 3.44 3.23 3.07 2.95 2.85 2.77 2.70 2.64 2.59 2.55 2.51 2.48 2.45 2.42 2.40 2.37 2.36 2.34 2.32 2.31 2.29 2.28 2.27 2.18 2.13 2.10 2.07 2.06 241 19.4 8.81 6.00 4.77 4.10 3.68 3.39 3.18 3.02 2.90 2.80 2.71 2.65 2.59 2.54 2.49 2.46 2.42 2.39 2.37 2.34 2.32 2.30 2.28 2.27 2.25 2.24 2.22 2.21 2.12 2.07 2.04 2.02 2.00 242 19.4 8.79 5.96 4.74 4.06 3.64 3.35 3.14 2.98 2.85 2.75 2.67 2.60 2.54 2.49 2.45 2.41 2.38 2.35 2.32 2.30 2.27 2.25 2.24 2.22 2.20 2.19 2.18 2.16 2.08 2.03 1.99 1.97 1.95 243 19.4 8.76 5.94 4.70 4.03 3.60 3.31 3.10 2.94 2.82 2.72 2.63 2.57 2.51 2.46 2.41 2.37 2.34 2.3 I 2.28 2.26 2.23 2.21 2.20 2.18 2.17 2.15 2.14 2.13 2.I34 1.99 1.95 1.93 1.91 244 19.4 8.74 5.91 4.68 4.00 3.57 3.28 3.07 2.91 2.79 2.69 2.60 2.53 2.48 2.42 2.38 2.34 2.31 2.28 2.25 2.23 2.20 2.18 2.16 2.15 2.13 2.12 2.10 2.09 2.00 i .95 1.93 1.89 1.88 245 19.4 8.73 5.89 4.66 3.98 3.55 3.26 2.05 2.89 2.76 2.66 2.58 2.51 2.45 2.40 2.35 2.31 2.28 2.25 2.22 2.20 2.18 2.15 2.14 2.12 2.10 2.09 2.08 2.06 1.97 1.92 1.89 1.86 1.84 1.83 1.82 1.72 245 19.4 8.71 5.87 4.64 3.96 3.53 3.24 3.03 2.86 2.74 2.64 2.55 2.48 2.42 2.37 2.33 2.29 2.26 2.22 2.20 2.17 2.15 2.13 2.11 2.09 2.08 2.06 2.05 2.04 1.95 1.89 1.86 1.84 1.82 1.80 1.79 1.69 246 19.4 8.70 5.86 4.62 3.94 3.51 3.22 3.01 2.85 2.72 2.62 2.53 2.46 2.40 2.35 2.3 i 2.27 2.23 2.20 2.18 2.15 2.13 2.11 2.09 2.07 2.06 2.04 2.03 2.01 1.92 1.87 1.84 1.81 1.79 1.78 1.77 1.67 246 19.4 8.69 5.84 4.60 3.92 3.49 3.20 2.99 2.83 2.70 2.60 2.51 2.44 2.38 2.33 2.29 2.25 2.21 2.18 2.16 2:13 2.11 2.09 2.07 2.05 2.04 2.02 2.01 1.99 1.90 1.85 1.82 I .79 1.77 1.76 1.75 1.64 247 19.4 8.68 5.83 4.59 3.91 3.48 3.19 2.97 2.81 2.69 2.58 2.50 2.43 2.37 2.32 2.27 2.23 2.20 2.17 2.14 2.11 2.09 2.07 2.05 2.03 2.02 2.00 1.99 1.98 1.89 1.83 1.80 1.77 1.75 1.74 1.73 1.62 247 19.4 8.67 5.82 4.58 3.90 3.47 3.17 2.96 2.80 2.67 2.57 2.48 2.41 2.35 2.30 2.26 2.22 2.18 2.15 2.12 2.10 2.07 2.05 2.04 2.02 2.00 1.99 1.97 1.96 1.87 1.81 1.78 1.75 1.73 1.72 1.71 1.60 248 19.4 8.67 5.8 I 4.57 3.88 3,46 3.16 2.95 2.78 2.66 2.56 2.47 2.40 2.34 2.29 2.24 2.20 2.17 2.14 2.11 2.08 2.06 2.04 2.02 .00 .99 ,97 .96 .95 .85 .80 .76 .74 .72 1.70 1.69 1.59 248 19.4 8.66 5.80 4.56 3.87 3.44 3.15 2.94 2.77 2,65 2.54 2.46 2.39 2.33 2.28 2.23 2.19 2.16 2.12 2.10 2.07 2.05 2.03 2.01 1.99 1.97 1.96 1.94 1.93 !.84 1.78 1.75 1.72 1.70 1.69 1.68 1.57 250 19.5 8.62 5.75 4.50 3.81 3.38 3.08 2.86 2.70 2.57 2.47 2.38 2.31 2.25 2.19 2.15 2.1 I 2.07 2.04 2.01 1.98 1.96 1.94 1.92 1.90 1.88 [.87 1.85 1.84 1.74 1.69 1.65 1.62 1.60 1.59 1.57 1.46 251 19.5 8.59 5.72 4.46 3.77 3.34 3.04 2.83 2.66 2.53 2.43 2.34 2.27 2.20 2.15 2.10 2.06 2.03 1.99 1.96 1.94 i .91 1.89 1.87 1.85 1.84 1.82 1.81 1.79 1.69 1.63 1.59 1.57 1.54 [.53 1.52 1.39 252 19.5 8.58 5.70 4.44 3.75 3.32 3.02 2.80 2.64 2.51 2.40 2.31 2.24 2.18 2.12 2.08 2.04 2.00 1.97 1.94 1.9! 1.88 1.86 1.84 1.82 1.81 [.79 1.77 1.76 1.66 1.60 1.56 1.53 1.51 1.49 1.48 1.35 252 19.5 8.57 5.69 4.43 3.74 3.30 3.01 2.79 2.62 2.49 2.38 2.30 2.22 2.16 2.11 2.06 2.02 1.98 1.95 1.92 1.89 1.86 1.84 1.82 1.80 1.79 1.77 1.75 1.74 1.64 1.58 1.53 1.50 1.48 1.46 1.45 1.32 252 19.5 8.56 5.67 4.41 3.72 3.29 2.99 2.77 2.60 2.47 2.36 2.27 2.20 2.14 2.08 2.03 1.99 1.96 1.92 1.89 1.86 1.84 1.82 1.80 1.78 1.76 1.74 1.73 1.71 1.61 1.54 1.50 1.47 1.45 1.43 1.41 1.27 253 19.5 8.55 5.66 4.41 3.71 3.27 2.97 2.76 2.59 2.46 2.35 2.26 2.19 2.12 2.07 2.02 i .98 1.94 1.91 .88 .85 .82 .80 .78 .16 .74 .73 .71 .70 1.59 1.52 148 1.45 1.43 [.41 1.39 1.24 254 19.5 8.53 5.63 4.37 3.67 3.23 2.93 2.71 2.54 2.40 2.30 2.21 2.13 2.07 2.01 1.96 1.92 1.88 1.84 181 1.78 1.76 1.73 1.7 I 1.69 1.67 1.65 1.64 1.62 1.51 i .44 1.39 1.35 1.32 1.30 1.28 1.00 ?0 11 w 12 13 14 15 16 17 ;: 20 21 ;: 24 25 :; 28 29 30 40 50 60 70 80 90 100 co NOTE -- (n, a ~ ~ y ~ e : ~ 0 2.47 2.32 2.20 3.95 3.10 2.71 2.46 2.31 2.19 3.94 3.09 2.70 2.37 2.21 2.10 3.84 3.00 2.60 ­ I) refers to the degrees of freedom for the 2.11 2.04 1.99 1.94 2.10 2.03 1.97 1.93 2.01 1.94 1.88 1.83 larger mean square placed in 1.90 1.86 1.89 1.85 1.79 1,75 the numerator. o w ANNEX E -- Continued CRITICAL VALUES OF THE F'-DISTR1BUTION Significance Degrees of Freedom (n, -I)+ (rl-1) .$ z ~ o 0 ~ Level 0.01 (For one-sided test) , 2 3 45678910 111213141516 171819203040 506080100 m": ~ b w o 0 u 1 2 3 4 5 6 7 8 9 4050 98.5 34. I 21.2 16.3 13.7 12.2 11.3 10.6 10.0 9.65 9.33 9.07 8.86 8.68 8.53 8.40 8.29 8.18 8.10 8.02 7.95 7,88 7.82 7.77 7.72 7.68 7.64 7.60 7.56 7.31 7.17 7.08 7.0 I 6.96 5000 99.0 30.8 18.0 I3.3 I0.9 9.55 8.65 8.02 7.56 7.2 I 6.93 6.70 6.51 6.36 6.23 6.11 6.01 5.93 5.85 5.78 5.72 5.66 5.61 5.57 5.53 5.49 5.45 5.42 5.39 5.18 5.06 4.98 4.92 4.88 5400 99.2 29,5 16.7 12.1 9.78 8.45 7'.59 6.99 6.55 6.22 5.95 5.74 5.56 5.42 5.29 5.18 5.09 5.01 4.94 4.87 4.82 4.76 4.72 4.68 4.64 4.60 4.57 4.54 4.51 4.31 4.20 4.13 4.08 4.04 5630 99.2 28.7 16.0 11.4 9.15 7.85 7.01 6.42 5.99 5.67 5.41 5.21 5.04 4.89 4.77 4.67 4.58 4.50 4.43 4.37 4.3 I 4.26 4.22 4.18 4.14 4.11 4.07 4.04 4.02 3.83 3.72 3.65 3.60 3.56 5760 99.3 28.2 15.5 11.0 8.75 7.46 6.63 6.06 5.64 5.32 5.06 4.86 4.70 4.56 4.44 4.34 4.25 4.17 4.10 4.04 3.99 3.94 3.90 3.86 3.82 3.78 3.75 3.73 3.72 3.51 3.41 3.34 3.29 3.26 5860 99.3 27.9 15.2 10.7 8.47 7.19 6.37 5.80 5.39 5.07 4.82 4.62 4.46 4.32 4.20 4.10 4.01 3.94 3.87 3.81 3.76 3.71 3.67 3.63 3.59 3.56 3.53 3.50 3.47 3.29 3.19 3.12 3.07 3.04 5930 99.4 27.7 15.0 10.5 8.26 6.99 6.18 5.61 5.20 4.89 4.64 4.44 4.28 4.14 4.03 3.93 3.84 3.77 3.70 3.64 3.59 3.54 3.50 3.46 3.42 3.39 3.36 3.33 3.30 3.12 3.02 2.95 2.91 2.87 5980 99.4 27.5 14.8 10.3 8.10 6.84 6.03 5.47 5.06 4,74 4.50 4.30 4.14 4.00 3.89 3.79 3.79 3.63 3.56 3.51 3.45 3.41 3.36 3.32 3.29 3.26 3.23 3.20 3.17 2.99 2.89 2.82 2.78 2.74 6020 99.4 27.3 14.7 I0.2 7.98 6.72 5.91 5.35 4.94 4.63 4.39 4.19 4.03 3.89 3.78 3.68 3.60 3.52 3.46 3.40 3.35 3.30 3.26 3.22 3,18 3.15 3.12 3.09 3.07 2.89 2.79 2.72 2.67 2.64 6060 99,4 27.2 14.5 10.1 7.87 6.62 5.81 5.26 4.85 4.54 4.30 4.10 3.94 3.80 3.69 3.59 3.5 I 3.43 3.37 3.31 3.26 3.21 3.17 3.13 3.09 3.06 3.o3 3.00 3.98 2.80 2.70 2.63 2.59 2.55 6080 99.4 27.1 14.4 9.96 7.79 6.54 5.73 5.18 4.77 4.46 4.22 4.02 3.86 3.73 3.62 3.52 3.43 3.36 3.29 3.24 3.18 3.14 3.09 3.06 3.02 2.99 2.96 2.93 2.91 2.73 2.63 2.56 2.51 2.48 6110 99.4 27.1 14.4 9.89 7.72 6.47 5.67 5.1 I 4.71 4.40 4.16 3.96 3.80 3.67 3.55 3.46 3.37 3.30 3.23 3.17 3.12 3.07 3.03 2.99 2.96 2.93 2.90 2.87 2.84 2.66 2.56 2.50 2.45 2.42 6130 99.4 27.0 14.3 9.85 7.66 6.41 5.61 5.05 4.65 4,34 4.10 3.91 3.75 3.61 3.50 3.40 3.32 3.24 3. [8 3.12 3.07 3.02 2.98 2.94 2.90 2.87 2.84 2.81 2.79 2.61 2.5 I 2.44 2.40 2.36 2.33 2.31 2.13 6140 99.4 26.9 14.2 9.77 7.60 6.36 5.56 5.00 4.60 4.29 4.05 3.86 3.70 3.56 3.45 3.35 3.27 3,19 3.13 3.07 3.02 2,97 2.93 2.89 2.86 2.82 2.79 2.77 2.74 2.56 2.46 2.39 2.35 2.31 2.29 2.26 2.08 6160 99.4 26.9 14.2 9.72 7.56 6.31 5.52 4.96 4.56 4,25 4.01 3.82 3.66 3.52 3.4 I 3.31 3.23 3.15 3.09 3.03 2.98 2.93 2.89 2.85 2.82 2.78 2.75 2.73 2.70 2.52 2.42 2.35 2.31 2.27 2.24 2.22 2.04 6170 99.4 26.8 14.2 9.68 7.52 6.27 5.48 4.92 4.52 4.21 3,97 3.78 3.62 3.49 3.37 3.27 3.19 3.12 3.05 2.99 2.94 2.89 2.85 2.81 2.78 2.75 2.72 2.69 2.66 2.48 2.38 2.31 2.27 2.23 2.21 2.19 2.00 6180 99.4 26.8 14.1 9.64 7.48 6.24 5.44 4.89 4.49 4.18 3.94 3.75 3.59 3.45 3.34 3.24 3.16 3.08 3.02 2.% 2.91 2.86 2.82 2.78 2.74 2.71 2.68 2.66 2.63 2.45 2.35 2.28 2.23 2.20 2.17 2.15 1.97 6190 99.4 26.8 14.1 9.61 7.45 6.21 5.41 4.86 4.46 4.15 3.91 3.72 3.56 3.42 3.31 3.21 3.13 3.05 2.99 2.93 2.88 2.83 2.79 2.75 2.72 2.68 2.65 2.63 2.60 2.42 2.32 2.25 2.20 2.17 2.14 2.12 1.93 6200 99.4 26.7 I4.0 9.5S 7.42 6.18 5.38 4.83 4.43 4.12 3.88 3.69 3.53 3.40 3.28 3.18 3.10 3.03 2.96 2.90 2.85 2.80 2.76 2.72 2.69 2.66 2.63 2.60 2.57 2.39 2.29 2.22 2.18 2.14 2.1 I 2.09 1.90 6210 99.4 26.7 14.0 9.55 7.40 6.16 5.36 4.81 4.41 4.10 3.86 3.66 3.5 I 3.37 3.26 3.16 3.08 3.00 2.94 2,88 2.83 2.78 2.74 2.70 2.66 2.63 2.60 2.57 2.55 2.37 2.27 2.20 2.15 2.12 2.09 2.07 1.88 6260 99.5 26.5 13.8 9.38 7.23 5.99 5.20 4.65 4.25 3.94 3.70 3.51 3.35 3,21 3.10 3.00 2.92 2.84 2.78 2.72 2.67 2.62 2.58 2.54 2.50 2.47 2.44 2.41 2.39 2.20 2.10 2.03 1.98 1.94 1.92 1.89 1.70 6290 99.5 26.4 13.7 9.29 7.14 5.91 5.12 4.57 4.17 3.86 3.62 3.43 3.27 3.13 3.02 2.92 2.84 2.76 2.69 2.64 2.58 2.54 2.49 2.45 2.42 2.38 2.35 2.33 2.30 2.11 2.01 1.94 1.89 1.85 1.82 1.80 1.59 6300 99.5 26.4 13.7 9.24 7.09 5.86 5.07 4.52 4.12 3.8 I 3.57 3.38 3.22 3.08 2.97 2.87 2.78 2.71 2.64 2.58 2.53 2.48 2.44 2.40 2.36 2.33 2.30 2.27 2.25 2.06 1.95 1.88 1.83 1.79 176 1.73 1.52 6310 99.5 26.3 13.7 9.20 7.06 5.82 5.03 4.48 4.08 3.78 3.54 3.34 3.18 3.05 2.93 2.83 2.75 2.67 2.61 2,55 2.50 2.45 2.40 2.36 2.33 2.29 2.26 2.23 2.21 2,02 1.91 1.84 1.78 1.75 1.72 1.69 1.47 6330 99.5 26.3 13.6 9.16 7.01 5.78 4.99 4.44 4.04 3.73 3.49 3.30 3.14 3.00 2.89 2.79 2.70 2.63 2.56 2.50 2.45 2.40 2.36 2.32 2.28 2.25 2.22 2.19 2.16 1.97 1.86 1.78 1.73 1.69 1.66 1.63 1.40 6330 99.5 26.2 13.6 9.13 6.99 5.75 4.96 4.42 4.01 3.71 3.47 3.27 3.1 I 2.98 2.86 2.76 2.68 2.60 2.54 2.48 2.42 2.37 2.33 2.29 2.25 2.22 2.19 2.16 2.13 1.94 1.82 1.75 1.70 i .66 1.62 1.60 1.36 6370 99.5 26.1 i3.5 9.02 6.88 5.65 4.86 4.3 I 3.91 3.60 3.36 3.17 3.00 2.87 2.75 2.65 2.57 2.49 2.42 2.36 2.31 2.26 2.21 2.17 2.13 2.10 2.06 2.03 2.01 1.80 1.68 1.60 1.54 1.49 1.46 1.43 1.00 10 11 12 13 14 15 16 17 18 ; 21 22 23 24 25 26 21 28 29 30 40 $ :: E 6.93 4.85 90 100 6.90 4.82 CO 6.63 4.61 NOlli -- (m ­ 1) reters to 4.0 I 3.54 3.23 3.01 2.84 2.72 2.61 2.52 3.98 3.51 3.2 I 2:99 2.82 2.69 2.59 2.50 3.78 3.32 3.02 2.80 2.64 2.5 I 2.41 2.32 the degrees of freedom fok the larger mean square placed in the 2.45 2.39 2,43 2.37 2.25 2.18 numcrstor. ANNEX E -- Continued CRITICAL VALUES OF THE F-DISTRIBUTION Significance Degrees of Level 0.05 (For two-sided test) Freedom (II-1)+ (nZ-l) J 1 2 : 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 -40 50 60 70 80 90 100 co ~ ~ 3 45678910 111213141516 171819203040 506080100m 648 38.5' 17.4 12.2 10.0 8.81 8.07 7.57 7.21 6.94 6.72 6.55 6.4 I 6.30 6.20 6.12 6.04 5.98 5.92 5.87 5.83 5.79 5.75 5.72 5.69 5.66 5.63 5.61 5.59 5.57 5.42 5.34 5.29 5.25 5.22 5.20 5.18 5.02 800 39.0 16.O 10.6 8.43 7.26 6.54 6.06 5.71 5.46 5.26 5.10 4.97 4.86 4.76 4.69 4.62 4.56 4.51 4.46 4.42 4.38 4.35 4.32 4.29 4.27 4.24 4.22 4.20 4.18 4.05 3.98 3.93 3.89 3.86 3.84 3.83 3.69 864 39.2 15.4 9.98 7.76 6.60 5.89 5.42 5.08 4.83 4.63 4.47 4.35 4.24 4.15 4.08 4.0 I 3.95 3.90 3.86 3.82 3.78 3.75 3.72 3.69 3.67 3.65 3.63 3.6 I 3.59 3.46 3.39 3.34 3.31 3.28 3.27 3.25 3.12 900 39.2 15.1 9.60 7.39 6.23 5.52 5.05 4.72 4.47 4.28 4.12 4.00 3.89 3.80 3.73 3.66 3.61 3.56 3.51 3.48 3.44 3.41 3.38 3.35 3.33 3.31 3.29 3.27 3.25 3.13 3.06 3.01 2.98 2.95 2.93 2.92 2.79 922 39.3 14.9 9.36 7.15 5,99 5.29 4.82 4.48 4.24 4.04 3.89 3.77 3.66 3.58 3.50 3.44 3.38 3.33 3.29 3.25 3.22 3.18 3.15 3.13 3.10 3.08 3.06 3.04 3.03 2.90 2.83 2.79 2.75 2.73 2.71 2.70 2.57 937 39.3 14.7 9.20 6.98 5.82 5.12 4.65 4.32 4.07 3.88 3.73 3.60 3.50 3.41 3.34 3.28 3.22 3.17 3.13 3.09 3.05 3.02 2.99 2.97 2.94 2.92 2.90 2.88 2.87 2.74 2.67 2.63 2..60 2.57 2.55 2.54 2.41 948 39.4 14.6 9.07 6.85 5.70 4.99 4.53 4.20 3.95 3.76 3.61 3.48 3.38 3.29 3.22 3.16 3.10 3.05 3.01 2.97 2.93 2.90 2.87 2.85 2.82 2.80 2.78 2.76 2.75 2.62 2.55 2,51 2.48 2.45 2.43 2.42 2.29 957 39.4 14.5 8.98 6.76 5.60 4.90 4.43 4.10 3.85 3.66 3.51 3.39 3.29 3.20 3.12 3.06 3.01 2.96 2.91 2.87 2.84 2.81 2.78 2.75 2.73 2.71 2.69 2.67 5.65 2.53 2.46 2.41 2.38 2.36 2.34 2.32 2.19 963 39.4 14.5 8.90 6.68 5.52 4.82 4.36 4.03 3.78 3.59 3.44 3.31 3.21 3.12 3.05 2.98 2.93 2.88 2.84 2.80 2.76 2.73 2.70 2.68 2.65 2.63 2.61 2.59 2.57 2.45 2.38 2.33 2.30 2.28 2.26 2.24 2. I 1 969 39.4 14.4 8.84 6.62 5.46 4.76 4.30 3.96 3.72 3.53 3.37 3.25 3.15 3.06 2.99 2.92 2.87 2.82 2.77 2.73 2.70 2.67 2.64 2.61 2.59 2.57 2.55 2.53 2.51 2.39 2.32 2.27 2.24 2.21 2. [9 2.18 2.05 973 39.4 14.4 8.79 6.57 5.41 4.71 4.24 3.91 3.66 3.47 3.32 3.20 3.09 3.01 2.93 2.87 2.81 2.76 2.72 2.68 2.65 2.62 2.59 2.56 2.54 2.51 2.49 2.48 2.46 2.33 2.26 2.22 2.18 2.16 2.14 2.12 1.99 977 39.4 14.3 8.75 6,52 5.37 4.67 4,20 3.87 3,62 3.43 3.28 3.15 3.05 2.96 2.89 2.82 2.77 2.72 2,68 2.64 2.60 2.57 2.54 2.51 2.49 2.47 2,45 2.43 2.41 2.29 2,22 2,17 2.14 2,1 I 209 2.08 1,94 980 39.4 14.3 8.72 6.49 5.33 4.63 4.16 3.83 3.58 3.39 3.24 3.12 3.01 2.92 2.85 2.79 2.73 2.68 2.64 2.60 2.56 2.53 2.50 2.48 2.45 2.43 2.41 2.39 2.37 2.25 2.18 2.13 2.10 2.07 2.05 2.04 1.90 983 39.4 14.3 8.69 6.46 5.30 4.60 4.13 3.80 3.55 3.36 3.21 3.08 2.98 2.89 2.82 2.75 2.70 2.65 2.60 2.56 2.53 2.50 2.47 2.44 2.42 2.39 2.37 2.36 2.34 2.21 2.14 2.09 2.06 2.03 2.02 2.00 1.87 985 39.4 14.3 8.66 6.43 5.27 4.57 4.10 3.77 3.52 3,33 3.18 3.05 2.95 2.86 2.79 2.72 2.67 2.62 2.57 2.53 2.50 2.47 2.44 2.41 2.39 2.36 2.34 2.32 2.31 2.18 2.1 I 2.06 2.03 2.00 1.98 1.97 1.83 987 39.4 14.2 8.64 6.41 5.25 4.54 4.08 3.74 3.50 3.30 3.15 3.03 2.92 2.S4 2.76 2.70 2.64 2.59 2.55 2.51 2.47 2.44 2.41 2.38 2.36 2.34 2.32 2.30 2.28 2.15 2.08 2.03 2.00 1.97 1.95 1.94 1.80 989 39.4 14.2 8.62 6.39 5.23 4.52 4.05 3.72 3.47 3.28 3.13 3.00 2.90 2.81 2.74 2.67 2.62 2.57 2.52 2.48 2.45 2.42 2.39 2.36 2.34 2.31 2.29 2.27 2.26 2.13 2.06 2.01 1.97 1,95 1.93 1.91 1.78 990 39.4 14.2 8.60 6.37 5.2 I 4.50 4.03 3.70 3.45 3.26 3.11 2.98 2.88 2.79 2.72 2.65 2.60 2.55 2.50 2.46 2.43 2.39 2.36 2.34 2.31 2.29 2.27 2.25 2.23 2.11 2.03 1.98 1.95 1.93 1.91 1.89 1.75 992 39.4 14.2 8.58 6,35 5.19 4.48 4,02 3.68 3.44 3,24 3.09 2.96 2.86 2.77 2.70 2.63 2.58 2.53 2.48 2,44 2.41 2.37 2.35 2.32 2.29 2.27 2.25 2.23 2.21 2.09 2.01 1.96 I .93 1.90 1.88 1.87 1.73 993 39.4 14.2 8.56 6.33 5.17 4.47 4.00 3.67 3.42 3.23 3.07 2.95 2.84 2.76 2.63 2.62 2.56 2.51 2.46 2.42 2.39 2.36 2.33 2.30 2.28 2.25 2.23 2.2 I 2.20 2.07 1.99 1.94 1.91 1.88 1.86 1.85 1.71 1001 39.5 14.1 8.46 6.23 5.07 4.36 3.89 3.56 3.31 3.12 2.96 2.84 2.73 2.64 2.57 2.50 2.44 2.39 2.35 2.3 I 2.27 2.24 2.21 2.18 2.16 2.13 2.11 2.09 2.07 1.94 1.87 1.82 1.78 1.75 1.73 1.71 1.57 1006 39.5 14.0 8.41 6.18 5.01 4.31. 3,84 3,51 3,26 3.06 2.91 2.78 2,67 2.58 2.51 2.44 2.38 2.33 2.29 2.25 2.21 2.18 2.15 2.12 2.09 2.07 2,05 2.03 2.01 1.88 1.80 1,74 1.71 1.68 1.66 1.64 1.48 1008 39.5 14.0 8.38 6.14 4.98 4.28 3.81 3.47 3.22 3.03 2.87 2.74 2.64 2.55 2.47 2.41 2.35 2.30 2.25 2.21 2.17 2.14 2.11 2.08 2.05 2.03 2.01 1.99 1.97 1.83 1.75 1.70 1.66 1.63 1.61 I.59 1.43 1010 39.5 14.0 8.36 6.12 4.96 4.25 3.78 3.45 3.20 3.00 2.85 2.72 2.61 2.52 2.45 2.38 2.32 2.27 2.22 2.18 2.14 2.11 2.08 2.05 2.03 2.00 1.98 1.96 1.94 1.80 1.72 1.67 1.63 1.60 1.58 1.56 1.39 1012 39.5 14.0 8.33 6.10 4.93 4.23 3.76 3.42 3.17 2.97 2.82 2.69 2.58 2.49 2.42 2.35 2.29 2.24 2.19 2.15 2.11 2.08 2.05 2.02 1.99 1.97 1.94 1.92 1.90 1.76 1.68 1.62 1.58 1.55 I.53 1.51 1.33 10I3 39.5 14.0 8.32 6.08 4.92 4.21 3.74 3.40 3.15 2.96 2.80 2.67 2.56 2.47 2.40 2.33 2.27 2.22 2.17 2.13 2.09 2.06 2.02 2.00 1,97 1.94 1.92 1.90 1.88 1.74 1.66 1.60 1.56 1.53 1.50 1.48 1.30 1018 39.5 13.9 8.26 6.02 4.85 4.14 3.67 3.33 3.08 2.88 2.72 2.60 2.49 2.40 2.32 2.25 2.19 2.13 2.09 2.04 2.00 1.97 1.94 1.91 1.88 1.85 1.83 1.81 1.79 1.64 1.55 1.48 1.44 1.40 1.37 1.35 1.00 NOTE -- (n, ­ I) refers to the degrees of freedom for the lar~er mean aouare r)laced in the numemtor. ANNEX E -- Concluded CRITICAL VALUES OF THE F-DISTRIBUTION Significance Degrees of Freedom (n, -I)+ (n*-l) , ~ 3 4567 S910 11121314151617 18192030405060 `0 100 m z m 0 : Level 0.01 (For two-sided test) ~ ~ + w U z u + 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 : 21 22 23 24 25 26 27 28 29 1620020000 198 199 55.6 49.8 31.3 26.3 22.8 18.3 18.6 16.2 14.7 13.6 14.5 12.4 I I.o 10.1 21600 199 47.5 24.3 16.5 12.9 I0.0 9.60 8.72 8.08 7.60 7.23 6.93 6.68 6.48 6.30 6.16 6.03 5.92 5.82 5.73 5.65 5.58 5.52 5.46 5.41 5.36 5.32 5.28 5.24 4.98 4.83 4.73 4.65 4.61 4.57 4.54 4.28 22500231002340023700 i 99 I99 199 199 46.2 45.4 44.8 44.4 23.2 22.5 22.0 21.6 15.6 14.9 14.5 14.2 12.0 10.0 8.81 7.96 7.34 6.88 6.52 6.23 6.00 5.80 5.64 5.50 5.37 5.27 5.17 11.5 9.52 8.30 7.47 6.87 6.42 6.07 5.79 5.56 5.37 5.21 5.07 4.96 4.85 4.76 11.1 9.16 7.95 7.13 6.54 6.10 5.76 5.48 5.26 5.07 4.91 4.78 4.66 4.56 4.47 10.8 8.89 7.69 6.88 6.30 5.86 5.52 5.25 5.03 4.85 4.69 4.56 4.44 4.34 4.26 23900241002420024300 I99 199 199 199 44.1 43.9 43.7 43.5 21.4 21.1 21.0 20.8 14.0 13.8 13.6 13.5 10.6 8.68 7.50 6.69 6.12 5.68 5.35 5.08 4.86 4.67 4.52 4.39 4.28 4.18 4.09 10.4 8.51 7.34 6.54 5.97 5.54 5.20 4.94 4.72 4.54 4.38 4.25 4.14 4.04 3.96 10.2 8.38 7.21 6.42 5.85 5.42 5.09 4.82 4.60 4.42 4.27 4.14 4.03 3.93 3.85 10. I 8.27 7.10 6.31 5.75 5.32 4.99 4.72 4.51 4.33 4.18 4.05 3.94 3.84 3.76 24400245002460024600 199 199 199 199 43.4 43.3 43.2 43.1 20.7 20.6 20.5 20.4 13.4 13.3 13.2 13.1 10.0 8.18 7.01 6.23 5.66 5.24 4.91 4.64 4.43 4.25 4.10 3.97 3.86 3.76 3.68 9.95 8.10 6.94 6.15 5.59 5.16 4.84 4.57 4.36 4.18 4.03 3.90 3.79 3.70 3.61 9.X8 8.03 6.87 6.09 5.53 5.10 4.77 4.51 4.30 4.12 3.97 3.84 3.73 3.64 3.55 9.81 7.97 6.81 6.03 5.47 5.05 4.72 4.46 4.25 4.07 3.92 3.79 3.68 3.59 3.50 24700247002480024800 199 199 199 19 43.0 42.9 42.9 42.i 20.4 20.3 20.3 20.2 13.1 13.0 13.0 12.9 9.76 7.93 6.76 5.98 5.42 5.00 4.67 4.41 4.20 4.02 3.87 3.75 3.64 3.54 3.46 9.71 7.87 6.72 5.94 5.38 4.96 4.63 4.37 4.16 3.98 3.83 3.71 3.60 3.50 3.42 9.66 7.83 6.68 5.90 5.34 4.92 4.59 4.33 4.12 3.95 3.80 3.67 3.56 3.46 3.38 9.62 7.79 6.64 5.86 5.30 4.89 4.56 4.30 4.09 3.91 3.76 3.64 3.53 3.43 3.35 24800250002510025200 100 i~.i 20.2 12.9 9.59 7.75 6.61 5.83 5.27 4.86 4.53 4.27 4.06 3.88 3.73 3.61 3.50 3.40 3.32 100 ii> 19.9 12.7 9.36 7.53 6.40 5.62 5.07 4.65 4.33 4.07 3.86 3.69 3.54 3.41 3.30 3.21 IQ'J ii: 19.8 12.5 9.21 7.42 6.29 5.52 4.97 4.55 4.23 3.97 3.76 3.5S 3.44 3.31 3.20 3.11 I 99 i22 19.7 12.5 9.17 7.35 6.22 5.45 4.90 4.49 4.17 3.91 3.70 3.52 3.37 3.25 3.14 3.04 253002530025300 I 99 I 99 I99 42.1 42.0 42.1 19.5 19.5 19.6 12.3 12.3 12.4 9.12 7.31 6.18 5.41 4.86 4.44 4.12 3.87 3.66 3.48 3.33 3.21 3.10 3.00 9.06 7.25 6.12 5.36 4.80 4.39 4.07 341 3.60 3.43 3.28 3.15 3.04 2.95 9.03 7.22 6.09 5.32 4.77 4.36 4.04 3.78 3.57 3.39 3.25 3.12 3.01 2.91 25500 200 41.8 19.3 12.1 8.88 7.08 5.95 5.19 4.64 4.23 3.90 3.65 3.44 3.26 3.11 2.98 2.87 2.78 12.s 12.2 11.8 11.4 11.1 10.8 10.6 10.4 10.2 10.1 9.94 9.83 9.73 9.63 9.55 9.48 9.41 9.34 9.28 9.23 9.18 8.83 8.63 8.49 8.40 8.33 8.28 8.24 7.88 9.43 8.91 S.51 8.19 7.92 7.70 7.51 7.35 7.21 7.09 6.99 6.89 6.81 6.73 6.66 6.60 6.54 6.49 6.44 6.40 6.35 6.07 5.90 5.80 5.72 5.67 5.62 5.59 5.30 & 3.12 3.05 2.98 2.92 2.87 2.82 "2.77 2.73 2.69 2.66 2.63 2.40 2.27 2.19 2.13 2.08 2.05 2.02 1.79 3.02 2.95 2.88 2.82 2.77 2.72 2.67 2.63 2.59 2.56 2.52 2.30 2.16 2.08 2.02 1.97 1.94 1.9 I 1.67 2.% 2.88 2.82 2.76 2.70 2.65 2.61 2.57 2.53 2.49 2.46 2.23 2.10 2.01 1.95 1.90 1.87 1.84 1.59 2.92 2.84 2.77 2.71 2.66 2.61 2.56 2.52 2.48 2.45 2.42 2.18 2.05 1.96 1.90 1.85 1.82 1.79 1.53 2.86 2.78 2.72 2.66 2.60 2.55 2.5 I 2.47 2.43 2.39 2.36 2.12 1.99 1.90 1.84 1.79 1.75 1.72 1.45 2.83 2.75 2.69 2.62 2.57 2.52 2.47 2.43 2.39 2.36 2.32 2.09 1.95 1.86 1.80 1.75 1.71 1.68 I.40 2.69 2.61 2.55 2.48 2.43 2.38 2.33 2.29 2.25 2.21 2.18 1.93 1.79 1.69 1.62 1.56 1.52 I .49 1.00 5.09 5.02 4.95 4.89 4.84 4.79 4.74 4.7o 4.66 4.62 4.37 4.23 4.!4 4.08 4.03 3.99 3.96 3.72 4.68 4.61 4.54 4.49 4.43 4.38 4.34 4.3o 4.26 4.23 3.99 3.85 3.76 3..70 3.65 3.62 3.59 3.35 4.39 4.32 4.26 4.20 4.15 4.10 4.06 4.02 3.98 3.95 3.71 3.58 3.49 3.43 3.39 3.35 3.33 3.09 4.18 4.11 4.05 3.99 3.94 3.89 3.85 3.81 3.77 3.74 3.51 3.38 3.29 3.23 3.19 3.15 3.13 2.90 4.01 3.94 3.88 3.83 3.78 3.73 3.69 3.65 3.61 3.58 3.35 3.22 3.13 3.08 3.03 3.00 2.97 2.74 3,88 3.81 3.75 3.69 3.64 3.60 3.56 3.52 3.48 3.45 3.22 3.09 3.01 2.95 2.91 2.87 2.85 2.62 3.77 3.70 3.64 3.59 3.54 3.49 3.45 3.41 3.38 3.34 3.12 2.99 2.90 2.85 2.80 2.77 2.74 2.52 3.68 3.61 3.55 3.50 3.45 3.40 3.36 3.32 3.29 3.25 3.03 2.90 2.82 2.76 2.72 2.68 2.66 2.43 3.60 3.54 3.47 3.42 3.37 3.33 3.28 3.25 3.21 3.18 2.95 2.82 2.74 2.68 2.64 2.61 2.58 2.36 3.54 3.47 3.41 3.35 3.30 3.26 3.22 3.18 3.15 3.11 2.89 2.76 2.68 2.62 2.58 2.54 2.52 2.29 3.48 3.41 3.35 3.30 3.25 3.20 3.16 3.12 3.09 3.06 2.83 2.70 2.62 2.56 2.52 2.49 2.46 2.24 3.43 3.36 3.30 3.25 3.20 3.15 3.1 I 3.07 3.04 3.01 2.78 2.65 2.57 2.51 2.47 2.44 2.41 2.19 3.38 3.31 3.25 3.20 3.15 3.11 3.07 3.03 2.99 2.% 2.74 2.61 2.53 2.47 2.43 2.39 2.37 2.14 3.34 3.27 3.21 3.16 3.1 I 3.07 3.03 2.99 2.95 2.92 2.70 2.57 2.49 2.43 2.39 2.35 2.33 2.10 3.31 3.24 3.18 3.12 3.08 3.03 2.99 2.9$ 2.92 2.89 2.66 2.53 2.45 2.39 2.35 2.32 2.29 2.06 3.27 3.20 3.15 3.09 3.04 3.@3 2.96 2.92 2.88 2.8S 2.63 2.50 2.42 2.36 2.32 2.28 2.26 2.03 3.24 3.18 3.12 3.06 3.01 2.97 2.93 2.89 2.86 2.82 2.60 2.47 2.39 2.33 2.29 2.25 2.23 2.00 30 40 : 70 80 90 100 NOTE -- (n! ­ I) refem to the demtxs of freedom for the larger mean square placedin the numerator. IS 6200 (Part 1) :2003 ANNEX F (Foreword) COMMITTEE COMPOSITION Statistical Methods for Quality and Reliability Sectional Committee, MSD 3 Organization Kolkata University, Kolkata Bharat Heavy Electrical Limited, Hyderabad Representative(s) PROF S. P. MUKHEFUEE (Chairman) SHRIS. N. JHA SHRIA. V. KRISHNAN (Alternate) Continental Devices India Ltd, New Delhi Directorate General of Quality Assurance, New Delhi Directorate of Standardization, Escorts Limited, Faridabad HMT Ltd, R&D Centre, Barrgalore Indian Agricultural Statistics Research Institute (lASRI), New Delhi Indian Association for Productivity, Indian Institute of Management Quality & Reliability (IAPQR), Kolkata Ministry of Defence, New Delhi SHRI G. V. SUBRAMANIAN SRIMATI RENUKAUL(Alfernate) SHRIS. K. SRtVASTVA LT COL P. VIJAYAN (Alternate) DR ASHOK KUMAR SHRJ C. S. V. NAKENDRA SHRI K. VIJAYAMMA DR S. D. SHARMA DR A. K. SRIVASTAVA (Alternate) DR B. DAS PROF S. CHAKRABORTV PROF. S. R. MOHAN PROF ARVINO SETH (Ahernale) (IIM), Lucknow Indian Statistical Institute (1S1), Kolkata National Institution for Quality and Reliability (NIQR), New Delhi Powergrid Corporation of India Ltd, New Delhi SRF Limited, Chennai Standardization, New Delhi Testing and Quality Certification Directorate (STQCD), SHRI Y. K. BHAT SHRJ G. W. DATEY(Ahernate) DR S. K. AGARWAL SHRID. CHAKRABORTV (Alternate) SHRI A. SANJEEVA No SHRt RAMANI SUBRAMANIAN (Alternate) SHRIS. K. KIMOTHI SHttt P. N. Swxww (Alternate) SHRJ S. KUMAR SHRIStwm SARUP (Alterna/e) Tata Engineering and Locomotive Co Ltd (TELCO), Jamshedpur University of Delhi, Delhi In personal capacity (20/1, Krishna Nagar, Safdarjung Enclave, New Delhi 110029) In personal capacity (B-109, Malviya Nagar, BIS Directorate General New Delhi 110017) PROF M. C. AGRAWAL SHRt D. R. SEN PROF A. N. NANKANA SHRI P. K. GAMBHIR, Director & Head (MSD) [Representing Member Secretary Shri Lalit Kumar Mehta Deputy Director (MSD), BIS Director General (Ex-o&io)] Basic Statistical Methods Subcommittee, MSD 3:1 Kolkata University, Kolkata Bajaj Auto Ltd, Pune Defence Science Centre, DRDO, Ministry of Defence, New Delhi Indian Agricultural Statistics Research Institute (IASRI), New Delhi Indian Association for Productivity, Quality and Reliability (IAPQR), Kolkata HM, Lucknow Indian Statistical Institute , New Delhi In personal capacity (B-109, Malviya Nagar, New Delhi 110017) In personal capacity (20/1, Krishna Nagar, Safdarjung Enclave, New Delhi 110029) (Continued on page 18) PROF S. P. MUKHERJEE (Convener) SHRI A. K. SRJVASTAVA DR ASHOK KUMAR DR S. D. SHARMA DR B. DAS DR A. LAHIRJ (Alternate) PROF. S. CHAKRA6GRTY PROFS. R. MOHAN PROFA. N. NANKANA SHRID. R. SEN 17 1S 6200 (Part 1) :2003 (Continned firm page 17) Panel for Basic Methods Including Terminology, MSD 3: VP-1 Organization NIQR, New Delhi Defence Science Centre, DRDO, Ministry of Defence, New Delhi Indian Agricultural Statistics Research Institute (IASRI), New Delhi Indian Statistical Institute (1S1), New Delhi National Institution for Quality and Reliability (NIQR), New Delhi Powergrid Corporation of India Ltd, New Delhi lnpersonal capacity (20/1, Krishna Nagar, Safdarjung Enclave, New Delhi 110029) Representative(s) SHSU G. W. DATEY(Convener) DR ASHOK KUMAR DR S. D. SHAItMA PKOF S. R. MOHAN SHIU Y. K. BHAT DR S. K. AGARWAL SHRID. R. SEN Bureau of Indian Standards is a statutory institution established under the Bureau of Indian Standards Act, 1986 to promote harmonious development of the activities of standardization, marking and quality certification of goods and attending to connected matters in the country. BIS Copyright BIS has the copyright of all its publications. 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